{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture9-new

Lecture9-new - 6.254 Game Theory with Engineering...

Info icon This preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
6.254 : Game Theory with Engineering Applications Lecture 9: Computation of NE in finite games Asu Ozdaglar MIT March 4, 2010 1
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Game Theory: Lecture 9 Introduction Introduction In this lecture, we study various approaches for the computation of mixed Nash equilibrium for finite games. Our focus will mainly be on two player finite games (i.e., bimatrix games). We will also mention extensions to games with multiple players and continuous strategy spaces at the end. The two survey papers [von Stengel 02] and [McKelvey and McLennan 96] provide good references for this topic. 2
Image of page 2
Game Theory: Lecture 9 Zero-Sum Finite Games Zero-Sum Finite Games We consider a zero-sum game where we have two players. Assume that player 1 has n actions and player 2 has m actions. We denote the n × m payoff matrices of player 1 and 2 by A and B . Let x denote the mixed strategy of player 1, i.e., x X , where X = { x | n i = 1 x i = 1, x i 0 } , and y denote the mixed strategy of player 2, i.e., y Y , where Y = { y | m j = 1 y j = 1, y j 0 } . Given a mixed strategy profile ( x , y ) , the payoffs of player 1 and player 2 can be expressed in terms of the payoff matrices as, u 1 ( x , y ) = x T Ay , u 2 ( x , y ) = x T By . 3
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Game Theory: Lecture 9 Zero-Sum Finite Games Zero-Sum Finite Games Recall the definition of a Nash equilibrium: A mixed strategy profile ( x * , y * ) is a mixed strategy Nash equilibrium if and only if ( x * ) T Ay * x T Ay * , for all x X , ( x * ) T By * ( x * ) T By , for all y Y . For zero-sum games, we have B = - A , hence the preceding relation becomes ( x * ) T Ay * ( x * ) T Ay , for all y Y . Combining the preceding, we obtain x T Ay * ( x * ) T Ay * ( x * ) T AY , for all x X , y Y , i.e., ( x * , y * ) is a saddle point of the function x T Ay defined over X × Y . Note that a vector ( x * , y * ) is a saddle point if x * X , y * Y , and sup x X x T Ay * = ( x * ) T Ay * = inf y Y ( x * ) T Ay . (1) 4
Image of page 4
Game Theory: Lecture 9 Zero-Sum Finite Games Zero-Sum Finite Games For any function φ : X × Y 7→ R , we have the minimax inequality : sup x X inf y Y φ ( x , y ) inf y Y sup x X φ ( x , y ) , (2) Proof: To see this, for every ¯ x X , write inf y Y φ ( ¯ x , y ) inf y Y sup x X φ ( x , y ) and take the supremum over ¯ x X of the left-hand side. Eq. (1) implies that inf y Y sup x X x T Ay sup x X x T Ay * = ( x * ) T Ay * = inf y Y ( x * ) T Ay sup x X inf y Y x T Ay , which combined with the minimax inequality [cf. Eq. (2)], proves that equality holds throughout in the preceding. Hence, a mixed strategy profile ( x * , y * ) is a Nash equilibrium if and only if ( x * ) T Ay * = inf y Y sup x X x T Ay = sup x X inf y Y x T Ay . We refer to ( x * ) T Ay * as the value of the game . 5
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Game Theory: Lecture 9 Zero-Sum Finite Games Zero-Sum Finite Games We next show that finding the mixed strategy Nash equilibrium strategies and the value of the game can be written as a pair of linear optimization problems.
Image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern